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Toric arrangements that come from graphsMarcelo Aguiar and Swee Hong Chan
Cornell University
Toric arrangements
Layman’s terms: lines on a donut.
Studied in connection to Kostant partition functions (DeConcini-Procesi ‘05), arithmetic matroids (Moci ‘12),arithmetic Tutte polynomial (D’Adderio-Moci ‘13), etc.
This talk is about toric arrangements that are built fromgraphs.
Motivation
The current study of toric graphic arrangements is mainlyfocused on the case of the standard torus.
We study graphic arrangements on two other types of tori,the coweight torus and the coroot torus.
We will see that these two arrangements tell us new thingsabout the acyclic orientations of the input graph.
ToriLet V be a real vector space.
A lattice L is the integer-span of a basis of V .The associated torus is the quotient T := V /L.
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Graphic arrangementsLet G be a simple connected graph.A(G ) is called the linear graphic arrangement.A(G ) is called the affine graphic arrangement.
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A(K3) A(K3)
Toric graphic arrangements, example 1
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L1 A(K3)
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Toric graphic arrangements, example 2
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L1
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Toric graphic arrangements, example 2
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A(K3)
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Toric graphic arrangements, example 2
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L2 A(K3)
A(K3, L2)
Root system of type A
Vn := {x ∈ Rn | x1 + · · ·+ xn = 0} (ambient space);
An−1 := {ei − ej | 1 ≤ i < j ≤ n} (root system of type A);
ZAn−1 := Z{1/n(e1 + · · ·+ en)− ei | 1 ≤ i ≤ n} ((co)weight lattice);
ZAn−1 := Z{ei − ej | 1 ≤ i < j ≤ n} ((co)root lattice).
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V3 ZA2 ZA2
One graph, three toric arrangements
The standard arrangement: V = Rn, L = Zn.
The coweight arrangement: V = Vn, L = ZAn−1.
The coroot arrangement: V = Vn, L = ZAn−1.
A(K3, ZA2) A(K3,ZA2)
Toric chambers
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A(K3, ZA2)
A(K3,ZA2)
Acyclic orientationsRecall the bijection of Greene and Zaslavsky (’83):
Chambers of A(G )↔ Acyclic orientations of G
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Coweight Voronoi cells
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Coweight Voronoi relation
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Two orientations that are projected to the same toric chamber areVoronoi equivalent.
Coweight Voronoi relation
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Two orientations that are projected to the same toric chamber areVoronoi equivalent.
Coroot Voronoi cells
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Coroot Voronoi relation
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No two distinct acyclic orientations are Voronoi equivalent.
Coroot Voronoi relation
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No two distinct acyclic orientations are Voronoi equivalent.
Combinatorial description for coweight Voronoi equivalence
The relation is source-to-sink flip.
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Studied by
Mosesjan (‘72) and Pretzel (‘86) in combinatorics;Eriksson and Eriksson (‘09), and Speyer (‘09) in connection toconjugacy of Coxeter elements;Develin, Macauley and Reiner (‘16) in the context of toricarrangements.
It also arises in connection to sandpile groups and chip-firing.
Combinatorial description for coroot Voronoi equivalence
The relation has several equivalent descriptions.
One is source-sink exchange.
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Another one is n-step source-to-sink flip.
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THANK YOU!
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